#7645421

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,36)(28,34)(29,31)(30,33)(32,35)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(25,27,32,30)(26,33,35,36)(28,39,29,37)(31,38,34,40);(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,28,32,29)(26,34,35,31)(27,37,30,39)(33,38,36,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,29,32,28)(26,31,35,34)(27,37,30,39)(33,38,36,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,32,27)(26,33,35,36)(28,39,29,37)(31,40,34,38);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,31)(26,28)(27,38)(29,35)(30,40)(32,34)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552617);